2018/09/04 by Andres F. Fontalvo Orozco, Orozco, Andres F. Fontalvo, Azat M. Gainutdinov +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1809.01122
openalex publication_date 2018/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a finite-dimensional pivotal and unimodular Hopf algebra over a field k. It was shown in [BBGa] that the projective tensor ideal in H-mod admits a unique non-degenerate modified trace, a natural generalisation of the categorical trace. This paper provides an extension of this result to a much more general setting. We first extend the notion of the modified trace to the so-called module trace for a given k-linear module category M over a pivotal category C equipped with a module endofunctor. We provide a non-trivial class of examples of such module traces. In particular, we show that any finite-dimensional pivotal Hopf k-algebra, not necessarily unimodular, admits a non-degenerate module trace on its projective tensor ideal. We also extend this result to pivotal Hopf group-coalgebras of finite type, and we give explicit calculations for the family of Taft Hopf algebras and non-restricted Borel quantum groups at roots of unity.